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<div class="subTitle">org.apache.commons.math3.transform</div>
<h2 title="Class FastHadamardTransformer" class="title">Class FastHadamardTransformer</h2>
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<pre>public class <span class="strong">FastHadamardTransformer</span>
extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a>
implements <a href="../../../../../org/apache/commons/math3/transform/RealTransformer.html" title="interface in org.apache.commons.math3.transform">RealTransformer</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/io/Serializable.html?is-external=true" title="class or interface in java.io">Serializable</a></pre>
<div class="block">Implements the <a href="http://www.archive.chipcenter.com/dsp/DSP000517F1.html">Fast Hadamard Transform</a> (FHT).
 Transformation of an input vector x to the output vector y.
 <p>
 In addition to transformation of real vectors, the Hadamard transform can
 transform integer vectors into integer vectors. However, this integer transform
 cannot be inverted directly. Due to a scaling factor it may lead to rational results.
 As an example, the inverse transform of integer vector (0, 1, 0, 1) is rational
 vector (1/2, -1/2, 0, 0).</div>
<dl><dt><span class="strong">Since:</span></dt>
  <dd>2.0</dd>
<dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../serialized-form.html#org.apache.commons.math3.transform.FastHadamardTransformer">Serialized Form</a></dd></dl>
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<th class="colOne" scope="col">Constructor and Description</th>
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<td class="colOne"><code><strong><a href="../../../../../org/apache/commons/math3/transform/FastHadamardTransformer.html#FastHadamardTransformer()">FastHadamardTransformer</a></strong>()</code>&nbsp;</td>
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<td class="colFirst"><code>protected double[]</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/transform/FastHadamardTransformer.html#fht(double[])">fht</a></strong>(double[]&nbsp;x)</code>
<div class="block">The FHT (Fast Hadamard Transformation) which uses only subtraction and
 addition.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>protected int[]</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/transform/FastHadamardTransformer.html#fht(int[])">fht</a></strong>(int[]&nbsp;x)</code>
<div class="block">Returns the forward transform of the specified integer data set.</div>
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<tr class="altColor">
<td class="colFirst"><code>double[]</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/transform/FastHadamardTransformer.html#transform(double[],%20org.apache.commons.math3.transform.TransformType)">transform</a></strong>(double[]&nbsp;f,
         <a href="../../../../../org/apache/commons/math3/transform/TransformType.html" title="enum in org.apache.commons.math3.transform">TransformType</a>&nbsp;type)</code>
<div class="block">Returns the (forward, inverse) transform of the specified real data set.</div>
</td>
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<tr class="rowColor">
<td class="colFirst"><code>int[]</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/transform/FastHadamardTransformer.html#transform(int[])">transform</a></strong>(int[]&nbsp;f)</code>
<div class="block">Returns the forward transform of the specified integer data set.The
 integer transform cannot be inverted directly, due to a scaling factor
 which may lead to double results.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double[]</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/transform/FastHadamardTransformer.html#transform(org.apache.commons.math3.analysis.UnivariateFunction,%20double,%20double,%20int,%20org.apache.commons.math3.transform.TransformType)">transform</a></strong>(<a href="../../../../../org/apache/commons/math3/analysis/UnivariateFunction.html" title="interface in org.apache.commons.math3.analysis">UnivariateFunction</a>&nbsp;f,
         double&nbsp;min,
         double&nbsp;max,
         int&nbsp;n,
         <a href="../../../../../org/apache/commons/math3/transform/TransformType.html" title="enum in org.apache.commons.math3.transform">TransformType</a>&nbsp;type)</code>
<div class="block">Returns the (forward, inverse) transform of the specified real function,
 sampled on the specified interval.</div>
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<code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#clone()" title="class or interface in java.lang">clone</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#equals(java.lang.Object)" title="class or interface in java.lang">equals</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#finalize()" title="class or interface in java.lang">finalize</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#getClass()" title="class or interface in java.lang">getClass</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#hashCode()" title="class or interface in java.lang">hashCode</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notify()" title="class or interface in java.lang">notify</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notifyAll()" title="class or interface in java.lang">notifyAll</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#toString()" title="class or interface in java.lang">toString</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait()" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long)" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long,%20int)" title="class or interface in java.lang">wait</a></code></li>
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<h4>FastHadamardTransformer</h4>
<pre>public&nbsp;FastHadamardTransformer()</pre>
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<pre>public&nbsp;double[]&nbsp;transform(double[]&nbsp;f,
                 <a href="../../../../../org/apache/commons/math3/transform/TransformType.html" title="enum in org.apache.commons.math3.transform">TransformType</a>&nbsp;type)</pre>
<div class="block">Returns the (forward, inverse) transform of the specified real data set.</div>
<dl>
<dt><strong>Specified by:</strong></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/transform/RealTransformer.html#transform(double[],%20org.apache.commons.math3.transform.TransformType)">transform</a></code>&nbsp;in interface&nbsp;<code><a href="../../../../../org/apache/commons/math3/transform/RealTransformer.html" title="interface in org.apache.commons.math3.transform">RealTransformer</a></code></dd>
<dt><span class="strong">Parameters:</span></dt><dd><code>f</code> - the real data array to be transformed (signal)</dd><dd><code>type</code> - the type of transform (forward, inverse) to be performed</dd>
<dt><span class="strong">Returns:</span></dt><dd>the real transformed array (spectrum)</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></code> - if the length of the data array is
 not a power of two</dd></dl>
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<pre>public&nbsp;double[]&nbsp;transform(<a href="../../../../../org/apache/commons/math3/analysis/UnivariateFunction.html" title="interface in org.apache.commons.math3.analysis">UnivariateFunction</a>&nbsp;f,
                 double&nbsp;min,
                 double&nbsp;max,
                 int&nbsp;n,
                 <a href="../../../../../org/apache/commons/math3/transform/TransformType.html" title="enum in org.apache.commons.math3.transform">TransformType</a>&nbsp;type)</pre>
<div class="block">Returns the (forward, inverse) transform of the specified real function,
 sampled on the specified interval.</div>
<dl>
<dt><strong>Specified by:</strong></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/transform/RealTransformer.html#transform(org.apache.commons.math3.analysis.UnivariateFunction,%20double,%20double,%20int,%20org.apache.commons.math3.transform.TransformType)">transform</a></code>&nbsp;in interface&nbsp;<code><a href="../../../../../org/apache/commons/math3/transform/RealTransformer.html" title="interface in org.apache.commons.math3.transform">RealTransformer</a></code></dd>
<dt><span class="strong">Parameters:</span></dt><dd><code>f</code> - the function to be sampled and transformed</dd><dd><code>min</code> - the (inclusive) lower bound for the interval</dd><dd><code>max</code> - the (exclusive) upper bound for the interval</dd><dd><code>n</code> - the number of sample points</dd><dd><code>type</code> - the type of transform (forward, inverse) to be performed</dd>
<dt><span class="strong">Returns:</span></dt><dd>the real transformed array</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NonMonotonicSequenceException.html" title="class in org.apache.commons.math3.exception">NonMonotonicSequenceException</a></code> - if the lower bound is greater than, or equal to the upper bound</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotStrictlyPositiveException.html" title="class in org.apache.commons.math3.exception">NotStrictlyPositiveException</a></code> - if the number of sample points is negative</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></code> - if the number of sample points is not a power of two</dd></dl>
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<pre>public&nbsp;int[]&nbsp;transform(int[]&nbsp;f)</pre>
<div class="block">Returns the forward transform of the specified integer data set.The
 integer transform cannot be inverted directly, due to a scaling factor
 which may lead to double results.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>f</code> - the integer data array to be transformed (signal)</dd>
<dt><span class="strong">Returns:</span></dt><dd>the integer transformed array (spectrum)</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></code> - if the length of the data array is not a power of two</dd></dl>
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<pre>protected&nbsp;double[]&nbsp;fht(double[]&nbsp;x)
                throws <a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></pre>
<div class="block">The FHT (Fast Hadamard Transformation) which uses only subtraction and
 addition. Requires <code>N * log2(N) = n * 2^n</code> additions.

 <h3>Short Table of manual calculation for N=8</h3>
 <ol>
 <li><b>x</b> is the input vector to be transformed,</li>
 <li><b>y</b> is the output vector (Fast Hadamard transform of <b>x</b>),</li>
 <li>a and b are helper rows.</li>
 </ol>
 <table align="center" border="1" cellpadding="3">
 <tbody align="center">
 <tr>
     <th>x</th>
     <th>a</th>
     <th>b</th>
     <th>y</th>
 </tr>
 <tr>
     <th>x<sub>0</sub></th>
     <td>a<sub>0</sub> = x<sub>0</sub> + x<sub>1</sub></td>
     <td>b<sub>0</sub> = a<sub>0</sub> + a<sub>1</sub></td>
     <td>y<sub>0</sub> = b<sub>0</sub >+ b<sub>1</sub></td>
 </tr>
 <tr>
     <th>x<sub>1</sub></th>
     <td>a<sub>1</sub> = x<sub>2</sub> + x<sub>3</sub></td>
     <td>b<sub>0</sub> = a<sub>2</sub> + a<sub>3</sub></td>
     <td>y<sub>0</sub> = b<sub>2</sub> + b<sub>3</sub></td>
 </tr>
 <tr>
     <th>x<sub>2</sub></th>
     <td>a<sub>2</sub> = x<sub>4</sub> + x<sub>5</sub></td>
     <td>b<sub>0</sub> = a<sub>4</sub> + a<sub>5</sub></td>
     <td>y<sub>0</sub> = b<sub>4</sub> + b<sub>5</sub></td>
 </tr>
 <tr>
     <th>x<sub>3</sub></th>
     <td>a<sub>3</sub> = x<sub>6</sub> + x<sub>7</sub></td>
     <td>b<sub>0</sub> = a<sub>6</sub> + a<sub>7</sub></td>
     <td>y<sub>0</sub> = b<sub>6</sub> + b<sub>7</sub></td>
 </tr>
 <tr>
     <th>x<sub>4</sub></th>
     <td>a<sub>0</sub> = x<sub>0</sub> - x<sub>1</sub></td>
     <td>b<sub>0</sub> = a<sub>0</sub> - a<sub>1</sub></td>
     <td>y<sub>0</sub> = b<sub>0</sub> - b<sub>1</sub></td>
 </tr>
 <tr>
     <th>x<sub>5</sub></th>
     <td>a<sub>1</sub> = x<sub>2</sub> - x<sub>3</sub></td>
     <td>b<sub>0</sub> = a<sub>2</sub> - a<sub>3</sub></td>
     <td>y<sub>0</sub> = b<sub>2</sub> - b<sub>3</sub></td>
 </tr>
 <tr>
     <th>x<sub>6</sub></th>
     <td>a<sub>2</sub> = x<sub>4</sub> - x<sub>5</sub></td>
     <td>b<sub>0</sub> = a<sub>4</sub> - a<sub>5</sub></td>
     <td>y<sub>0</sub> = b<sub>4</sub> - b<sub>5</sub></td>
 </tr>
 <tr>
     <th>x<sub>7</sub></th>
     <td>a<sub>3</sub> = x<sub>6</sub> - x<sub>7</sub></td>
     <td>b<sub>0</sub> = a<sub>6</sub> - a<sub>7</sub></td>
     <td>y<sub>0</sub> = b<sub>6</sub> - b<sub>7</sub></td>
 </tr>
 </tbody>
 </table>

 <h3>How it works</h3>
 <ol>
 <li>Construct a matrix with <code>N</code> rows and <code>n + 1</code> columns,
 <code>hadm[n+1][N]</code>.<br/>
 <em>(If I use [x][y] it always means [row-offset][column-offset] of a
 Matrix with n rows and m columns. Its entries go from M[0][0]
 to M[n][N])</em></li>
 <li>Place the input vector <code>x[N]</code> in the first column of the
 matrix <code>hadm</code>.</li>
 <li>The entries of the submatrix <code>D_top</code> are calculated as follows
     <ul>
         <li><code>D_top</code> goes from entry <code>[0][1]</code> to
         <code>[N / 2 - 1][n + 1]</code>,</li>
         <li>the columns of <code>D_top</code> are the pairwise mutually
         exclusive sums of the previous column.</li>
     </ul>
 </li>
 <li>The entries of the submatrix <code>D_bottom</code> are calculated as
 follows
     <ul>
         <li><code>D_bottom</code> goes from entry <code>[N / 2][1]</code> to
         <code>[N][n + 1]</code>,</li>
         <li>the columns of <code>D_bottom</code> are the pairwise differences
         of the previous column.</li>
     </ul>
 </li>
 <li>The consputation of <code>D_top</code> and <code>D_bottom</code> are best
 understood with the above example (for <code>N = 8</code>).
 <li>The output vector <code>y</code> is now in the last column of
 <code>hadm</code>.</li>
 <li><em>Algorithm from <a href="http://www.archive.chipcenter.com/dsp/DSP000517F1.html">chipcenter</a>.</em></li>
 </ol>
 <h3>Visually</h3>
 <table border="1" align="center" cellpadding="3">
 <tbody align="center">
 <tr>
     <td></td><th>0</th><th>1</th><th>2</th><th>3</th>
     <th>&hellip;</th>
     <th>n + 1</th>
 </tr>
 <tr>
     <th>0</th>
     <td>x<sub>0</sub></td>
     <td colspan="5" rowspan="5" align="center" valign="middle">
         &uarr;<br/>
         &larr; D<sub>top</sub> &rarr;<br/>
         &darr;
     </td>
 </tr>
 <tr><th>1</th><td>x<sub>1</sub></td></tr>
 <tr><th>2</th><td>x<sub>2</sub></td></tr>
 <tr><th>&hellip;</th><td>&hellip;</td></tr>
 <tr><th>N / 2 - 1</th><td>x<sub>N/2-1</sub></td></tr>
 <tr>
     <th>N / 2</th>
     <td>x<sub>N/2</sub></td>
     <td colspan="5" rowspan="5" align="center" valign="middle">
         &uarr;<br/>
         &larr; D<sub>bottom</sub> &rarr;<br/>
         &darr;
     </td>
 </tr>
 <tr><th>N / 2 + 1</th><td>x<sub>N/2+1</sub></td></tr>
 <tr><th>N / 2 + 2</th><td>x<sub>N/2+2</sub></td></tr>
 <tr><th>&hellip;</th><td>&hellip;</td></tr>
 <tr><th>N</th><td>x<sub>N</sub></td></tr>
 </tbody>
 </table></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - the real data array to be transformed</dd>
<dt><span class="strong">Returns:</span></dt><dd>the real transformed array, <code>y</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></code> - if the length of the data array is not a power of two</dd></dl>
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<h4>fht</h4>
<pre>protected&nbsp;int[]&nbsp;fht(int[]&nbsp;x)
             throws <a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></pre>
<div class="block">Returns the forward transform of the specified integer data set. The FHT
 (Fast Hadamard Transform) uses only subtraction and addition.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - the integer data array to be transformed</dd>
<dt><span class="strong">Returns:</span></dt><dd>the integer transformed array, <code>y</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></code> - if the length of the data array is not a power of two</dd></dl>
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